2412 ieee transactions on mobile computing, vol....

Frequency Domain Packet Schedulingwith Stability Analysis for 3GPP LTE Uplink

Fengyuan Ren, Member, IEEE, Yinsheng Xu, Hongkun Yang,

Jiao Zhang, and Chuang Lin, Senior Member, IEEE

Abstract—In this paper, we investigate the Frequency Domain Packet Scheduling (FDPS) problem for 3GPP Long Term Evolution

(LTE) Uplink (UL). Instead of studying a specific scheduling policy, we provide a unified approach to tackle this issue. First, we

formalize a general LTE UL FDPS problem, which is suitable for various scheduling policies. Then, we prove that the problem is MAX

SNP-hard, which implies that approximation algorithms with constant approximation ratios are the best that we can hope for.

Therefore, we design two approximation algorithms, both of which have polynomial runtime. The first algorithm is based on a simple

greedy method. The second one is based on the Local Ratio (L-R) technique and it can approximately solve the LTE UL FDPS problem

with an approximation ratio of 2. To further analyze the stability of the 2-approximation L-R algorithm, we derive a specific FDPS

problem, which incorporates the queue length and channel quality information. We utilize the Lyapunov Drift to prove the L-R algorithm

is stable for any ð!0; �0Þ-admissible LTE UL systems. The simulation results indicate good performance of the L-R scheduler.

Index Terms—Long term evolution (LTE), uplink (UL), frequency domain packet scheduling (FDPS), Optimization Algorithm,

approximation ratio, local ratio, stability analysis, Lyapunov drift, ð!; �Þ-admissible

Ç

1 INTRODUCTION

THE Third Generation Partnership Project (3GPP) Long

Term Evolution (LTE) standardization is the nextforward step in cellular network services. The objective of

LTE is to achieve a high peak-data-rate that scales with

scalable system bandwidths, system capacity and cover-

age improvements, spectrum efficiency, latency reduction,

and packet optimized radio access [2]. An architecture of

LTE network is depicted in Fig. 1. Its functionality is

divided into three main domains: User Equipment (UE),

Evolved UMTS Terrestrial Radio Access Network (EU-TRAN), and System Architecture Evolution (SAE) core,

also known as Evolved Packet Core (EPC). EUTRAN,

which consists of enodeBs, is similar in function to the

combination of nodeB and radio network controller (RNC)

in the traditional UTRAN. This simplification is beneficial

to reduce the latency of all radio interface operations. The

eNodeBs are connected by the X2 interface, and they

connect to EPC networks using the S1 interface. The EPCnetwork serves as the equivalent GPRS networks via three

components, i.e., Mobility Management Entity (MME),

Serving Gateway (SGW), and PDN Gateway (PGW). The

MME provides tracking and paging for idle mode UEs.

The SGW provides switching and routing for user data

packets. The PGW provides access to external networks,such as the Internet.

Because of the robustness against multipath fading,higher spectral efficiency, and bandwidth scalability, theOrthogonal Frequency Division Multiplexing Access(OFDMA) has been selected for the LTE Downlink (DL)[3]. However, OFDM has a high Peak-to-Average PowerRatio (PAPR), which is undesirable and inefficient for theUE terminal and drains the battery very fast [4], [5]. Thusa precoded version of OFDM, the Single-Carrier FrequencyDivision Multiplexing Access (SC-FDMA) is selected forthe LTE Uplink (UL) [5], [6], [7]. SC-FDMA retains themultipath resistance and flexible subcarrier frequencyallocation offered by OFDM. Moreover, it has a signifi-cantly low PAPR like traditional single-carrier formatssuch as GSM, which compensates for the drawback withnormal OFDM. For the UL, LTE provides the user speedup to 50 Mbps in a 20-MHz channel [5].

Both in the LTE DL and UL, the system bandwidth isdivided into separable chunks denoted as Resource Blocks(RBs) (see Fig. 1). An RB is considered as the minimumscheduling resolution in the time-frequency domain. TheFrequency Domain Packet Scheduling (FDPS) allocatesdifferent RBs to individual users according to their currentchannel conditions, queue lengths, and other information.The FDPS policy is conducted during each TransmissionTime Interval (TTI, in LTE, 1TTI = 1 ms). The UL SC-FDMAis achieved by the FDPS assignment of different frequencyportions of bandwidth, which simultaneously realizesfrequency-domain multiplexing in concert with time-do-main scheduling.

The FDPS algorithm plays an important role for thesystem performance of LTE. Unfortunately, the excellentscheduling policies in conventional cellular networks can behardly adopted in the LTE UL system. Most of all, SC-FDMA

2412 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 12, NO. 12, DECEMBER 2013

. F. Ren, Y. Xu, J. Zhang, and C. Lin are with the Tsinghua NationalLaboratory for Information Science and Technology, Department ofComputer Science and Technology, Tsinghua University, FIT Building,Beijing 100084, China.E-mail: {renfy, xuysh08, zhangjiao08, clin}@csnet1.cs.tsinghua.edu.cn.

. H. Yang is with the Department of Computer Science, The University ofTexas at Austin, Austin, TX 78712. E-mail: [email protected].

Manuscript received 7 May 2011; revised 8 May 2012; accepted 22 Sept. 2012;published online 23 Oct. 2012.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number TMC-2011-05-0237.Digital Object Identifier no. 10.1109/TMC.2012.223.

1536-1233/13/$31.00 � 2013 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS

induces the continuous allocation constraint for the FDPS [8].Namely, the LTE UL FDPS requires all the RBs allocated to asingle user must be contiguous in frequency domain withineach TTI (see the Uplink in Fig. 1), because the underlyingwaveform of SC-FDMA is essentially single carrier [6].Hence, it becomes harder to perform appropriate resourcescheduling for the LTE uplink.

The LTE UL FDPS problem has been addressed inexisting literature. Lim et al. [9] propose two utility-basedscheduling schemes for SC-FDMA systems. Afterwards,this work was improved by taking the delayed channel stateinformation into consideration [10]. Ruiz de Temino et al.[11] propose three channel-aware scheduling algorithms toaddress the localized RB allocation. Lee et al. [12] formalizethe proportional-fair (PF) FDPS problem (PF-FDPS) for theLTE uplink as a combinatorial optimization problem andprove that the problem is NP-hard. They propose fourapproximation algorithms based on heuristics and conductan extensive simulation study on the performance of thefour heuristic algorithms. The performance of three distinctschedulers for LTE uplink are compared in [13]. However, itmainly focuses on the impact of flow-level dynamicsresulting from the random user behavior.

However, the selection of the scheduling policy for aspecific LTE UL system depends on a case by case analysis,which is not the focus of our work. In this paper, rather thana particular scheduling policy, we present a unifiedapproach to tackle the LTE UL FDPS problem. We definethe profit function that indicates the profit gained byallocating a set of contiguous RBs to an active user. Theprofit function is capable of expressing various schedulingobjectives, including the PF metric [12] and schedulingpolicies that combine utility maximization and queuestability [14]. Based on the profit function, we formalize ageneral FDPS problem for the LTE UL that can cover manyLTE UL scheduling objectives.

Furthermore, we address the hardness of the LTE ULFDPS problem. We prove that the scheduling problem isMAX SNP-hard, which means that the problem is improb-able to efficiently approximate within a certain ratio.Accordingly, the approximation algorithms with constantapproximation ratios are the best solutions that we can hopefor. Subsequently, we design two approximation algorithmsin company with provable approximation ratios, whichsolve the LTE UL FDPS problem in polynomial runtime.

The first algorithm is based on a greedy method. It isintuitive and easy to follow, but its approximation ratio isbounded by a slowly increasing function of the number ofactive users. The second one is designed on the basis of theLocal Ratio technique [15], and is called L-R Algorithmhereafter. Although the L-R Algorithm is relatively sophis-ticated, it achieves a constant approximation ratio of 2.

The assumption of an infinitely backlogged model inscheduling analysis is conventional in many literatures,which implies there are always arrival packets to serve.However, this is not always the case in practical systems. Inreal networks, packets are generated for each user inaccordance with a stochastic arrival process with respect toapplications. Andrews [16] makes a survey of schedulingtheory in wireless networks with system stability considera-tions in cases of infinite and finite backlog. In the follow-onwork [14], Andrews and Zhang analyze the multicarriersituation, where finite queues are fed by an admissiblearrival process, and point out the PF scheduler does notwork so well. In particular, it can result in the instability ofqueues [17]. Provided that the schedulers have no idea of thequeues length, they may even schedule an empty queuethough its channel quality is good. Meanwhile, the queuefull of packets may have no chance to send once it faces a badchannel condition. In those two unfavorable situations, thequeues are not stable in a bounded length. Therefore, it ispreferable to take system utility maximization and queuingstability into account simultaneously when we designscheduling algorithms. Besides, the well-known MaxWeightalgorithm is generalized from the single-carrier settings toaccommodate a number of natural optimization problems inthe multicarrier settings [14]. Moreover, the hardness ofthese problems are stated and the simple algorithmicsolutions are designed with provable performance bounds,which exactly keeps the queuing system stable.

Since the L-R Algorithm achieves a fairly good perfor-mance, it is worthy to investigate the stability of thealgorithm. We specify the profit function to incorporate thequeue length and channel quality information, and thusformalize a specific LTE UL FDPS problem. Because thespecific problem is a special case of the general problem, theL-R Algorithm can also be applied with the same approx-imation ratio of 2. Similar to the methodology used in [14],we utilize the Lyapunov Drift to prove the queuing stabilityfor the L-R Algorithm in case of finite buffer and arbitraryarrival process. First, we extend the definition of ð!; �Þ-admissible system to the LTE UL multi-carrier scenario. Theð!; �Þ-admissible explicitly describes the basic propositionsof some LTE UL systems, which will be stably scheduled bythe feasible FDPS algorithms. We assume that the optimalscheduler for LTE UL FDPS is stable for any ð!; �Þ-admissible LTE UL systems. Then, the Lyapunov drift iscomputed and the queuing stability of the L-R Algorithm isproved in any ð!0; �0Þ-admissible LTE UL systems. Finally,The values of !0 and �0 are calculated.

The remainder of the paper is organized as follows:Section 2 introduces the necessary background in the theoryof approximation and complexity. Section 3 gives thesystem model we study and formalizes the LTE UL FDPSproblem. Section 4 proves the LTE UL FDPS problem is

REN ET AL.: FREQUENCY DOMAIN PACKET SCHEDULING WITH STABILITY ANALYSIS FOR 3GPP LTE UPLINK 2413

Fig. 1. The architecture of 3GPP LTE network.

MAX SNP-hard and shows polynomial time approximationalgorithms with guaranteed approximation ratios arenecessary for practical LTE UL systems. In Section 5 and6, respectively, we give two approximation algorithmscomputable in polynomial time and find out their approx-imation ratios. Section 7 analyzes the queuing stability ofthe 2-approximation L-R Algorithm. The performance of theL-R Algorithm is evaluated in Section 8. Finally, we concludethe paper in Section 9.

2 PRELIMINARIES

In this section, we provide a brief introduction to the theoryof both optimization and stability used in this work.

2.1 Approximation Ratio of an ApproximationAlgorithm

In practice, many optimization problems are NP-hard,which means that exact solutions of these problems arewidely believed to be time consuming. This necessitiesefficient approximation algorithms that always computesolutions close to the optimum. However, different approx-imation algorithms can achieve different degrees ofapproximation for the same problem. So, we introduce thedefinition of approximation ratio to qualify the degree ofapproximation achieved by an approximation algorithm.

Assume that G is an instance of a maximizationproblem.1 We denote the size of its input by jGj and itsoptimal value by OPT ðGÞ. Let ALG be an approximationalgorithm for the maximization problem. For instance G, wedenote the value of ALG by ALGðGÞ. We say that ALG hasan approximation ratio of �ðjGjÞ [18] if, for any instance G,OPT ðGÞ is within a factor of �ðjGjÞ of ALGðGÞ:

OPT ðGÞ � �ðjGjÞ �ALGðGÞ:

We also call ALG a �ðjGjÞ-approximation algorithm. Whenthe approximation ratio is independent of the input size jGj,we will use the terms approximation ratio of � and �-approximation algorithm, indicating no dependence on jGj.

Note that �ðjGjÞ (or �) always � 1, and a smaller value of�ðjGjÞ (or �) indicates that the approximation algorithm hasa better performance in a worst-case sense. In particular,when � ¼ 1, the approximation algorithm ALG essentiallyfinds the optimal solution for any instance G.

2.2 Polynomial-Time Approximation Scheme(PTAS)

A polynomial-time approximation scheme [18] for amaximization problem is an approximation algorithm thattakes as an input not only an instance of the problem, butalso a value � > 0 such that for any fixed �, the scheme is að1þ �Þ-approximation algorithm, which is computable inpolynomial time in the size of the input instance.

In a technical sense, a PTAS is the best one can hope foran NP-hard optimization problem, assuming P 6¼ NP .

2.3 L-Reduction

Suppose that A and B are maximization problems. AnL-reduction [19] from A to B is a pair of functions R and S,

both computable in polynomial time, with the followingtwo additional properties:

First, if X is an instance of A with optimum OPT ðXÞ,then RðXÞ is an instance of B with optimum OPT ðRðXÞÞthat satisfies

OPT ðRðXÞÞ � � �OPT ðXÞ; ð1Þ

where � is a positive constant.Second, if s is any feasible solution of RðXÞ, then SðsÞ is a

feasible solution of X such that

OPT ðXÞ � VALðSðsÞÞ � � � ðOPT ðRðXÞÞ � V ALðsÞÞ; ð2Þ

where � is another positive constant particular to thereduction and VAL denotes the value of the feasiblesolution in both instances. Equation (2) guarantees that Sreturns a feasible solution of X, which is not much moresuboptimal than the given solution of RðXÞ. In particular, ifs is the optimal solution of RðXÞ, then SðsÞ is the optimalsolution of X.

L-reductions have the composition property [19]:

Lemma 1. If ðR;SÞ is an L-reduction from problem A to problemB, and ðR0; S0Þ is an L-reduction from problem B to problem C,then their composition ðR �R0; S0 � SÞ is an L-reduction fromA to C.

2.4 MAX-SNP Hardness

In computational complexity theory, SNP (from Strict NP) isa complexity class containing a limited subset of NP basedon its logical characterization in terms of graph-theoreticalproperties. The class MAX SNP is a subset of optimizationproblems derived from SNP. The formal definition of MAXSNP can be found in [19]. A problem is said to be MAXSNP-hard if all MAX SNP problems can be L-reduced tothis problem. MAX SNP-hard problems are hard toapproximate. It is shown in [19] that

Lemma 2. Any MAX SNP-hard problem does not have a PTASunless P ¼ NP .

Suppose A is a known MAX SNP problem, thus all theMAX SNP problems can be L-reduced to A. Once B can beL-reduced fromA, according to the composition property inLemma 1, all the MAX SNP problems can also be L-reducedto B, which indicates B is MAX SNP-Hard. In other words,to prove that a problem B is MAX SNP-hard, it suffices topresent an L-reduction from a known MAX SNP-hardproblem A to B.

2.5 Job Interval Selection Problem with k Intervalsper Job

The job interval selection problem with k intervals per job(JISPk) is stated as follows [20]:

Input. We are given n job, each of which is associatedwith k intervals on the real line. Thus, we have k � nintervals. For each interval l a starting time sl and afinishing time flð>slÞ is known, l ¼ 1; . . . ; k � n. All startingand finishing times are integers. An interval is said to beactive at time t if and only if t 2 ½sl; flÞ. Two intervalintersect if and only if there is a time t during which bothintervals are active.


1. We focus on maximization problems in this paper because LTE ULFDPS is a maximization problem.

Goal. Select as many intervals as possible such that 1) foreach job, at most one interval is selected from its associatedk intervals, and 2) no two selected intervals intersect.

Measure. The number of intervals selected.Spieksma [20] presents an L-reduction from MAX-

3SAT-B to JISP2. Since MAX-3SAT-B is MAX SNP-hard[19], according to Lemma 1, [20] essentially proves that

Lemma 3. JISP2 is MAX SNP-hard, and hence it does not have a

PTAS unless P ¼ NP .

In the JISPk, k is a parameter of problem. In this paper,we refer to the JISP2 problem, namely, k ¼ 2.

2.6 Local Ratio for Scheduling Problems

The local ratio technique is a methodology for the designand analysis of algorithms in a broad range of optimizationproblems. It is first developed by Bafna et al. [21], laterextended by Bar-Yehuda [22], both of which treat mini-mization covering problems. Bar-Noy et al. [15] present thefirst local ratio algorithm for maximization problems. Bar-Yehuda and Rawitz, [23] develop a novel extension of thetechnique, called emphfractional local ratio.

Local ratio can be utilized to solve the profit maximiza-tion problem, that is, given a profit vector p 2 IRn, to find asolution vector x that maximizes the inner product p � x,subject to a given set IF of feasibility constraints on x. Thefundamental Local Ratio Theorem for maximization pro-blems is presented next, with proof omitted.

Theorem 1 (Local ratio). Let IF be a set of constraints and let

p, p1, and p2 be profit vectors such that p ¼ p1 þ p2. Then, if

x is an r-approximation solution with respect to ðIF; p1Þ and

with respect to ðIF; p2Þ, it is an r-approximation solution with

respect to ðIF; pÞ.

A massive applications of the technique are solvingthe Machine Scheduling problem. The resource consists ofk parallel machines and the activities are jobs to bescheduled on these machines. Each job can be scheduledin one of several time intervals. The goal is to maximizethe profit of the executed jobs. There are several subcasesto this problem, including Interval Scheduling, Indepen-dent Set in Interval Graphs, k-Colorable Subgraph,Parallel Unrelated Machines, and Scheduling with ReleaseTimes and Deadlines. Besides, the technique can be usedfor Bandwidth Allocation. The problem aims at finding themost profitable set of sessions that can utilize theavailable bandwidth. A 5-approximation algorithm basedon local ratio is designed to tackle it. The above problemsare all for profit maximization, which is similar to theLTE UL FDPS problem that we focus in this paper. Dueto the page limits, other applications of local ratio, forexample, the Loss Minimization and the detailed surveycan be found in [24].

2.7 Lyapunov Stability Analysis

According to [25], the definition of system stability and itssufficient conditions are presented. Suppose the state spaceis partitioned in the sets T;R1; R2 . . . , where Rj; j ¼ 1; 2 . . . ,are closed sets of communicating states and T contains all

states that do not belong to any closed set of communicatingstates and therefore are transient.

Definition 1. The system is stable if for the queue length

process Y, we have

P ð�y <1Þ ¼ 1 8y 2 T;

where �y is the recurrent time and all states y 2 [1j¼1Rj are

positive recurrent.

Theorem 2 states the sufficient conditions for systemstability based on Definition 1. Those conditions involvethe drift of a Lyapunov function on the state space of aMarkov chain.

Theorem 2. Consider a Markov chain MðtÞ with state spaceM.

If there exists a lower bounded real function V :M! IR,

� > 0 and a finite subset M0 of M such that

E½V ðMðtþ 1ÞÞ � V ðMðtÞÞ jMðtÞ ¼ y� � �� if y 62 M0

E½V ðMðtþ 1ÞÞ jMðtÞ ¼ y� <1 if y 2 M0;

then for the time �y in Definition 1, we have

P ð�y <1Þ ¼ 1 8y 2 T;

and all states y 2 [1j¼1Rj are positive recurrent.

The detailed introduction and formal definition of theLyapunov stability theory can be found in [26].

3 PROBLEM FORMALIZATION

3.1 System Model

We consider the uplink of cellular network, where thesystem bandwidth is divided into m RBs. Besides, thenetwork has a single base station and n active wirelessusers. We denote the set of all RBs by M (M ¼ f1; 2; . . . ;mg)and the set of all users by N (N ¼ f1; 2; . . . ; ng). Within eachTTI, the FDPS allocates m RBs to n users in a contiguous

fashion. Namely, a set of contiguous RBs is distributed toeach user while the empty set of RBs to a user means he isnot scheduled in this round. Specifically, each RB isassigned to at most one user. Fig. 2 illustrates an exampleof a feasible FDPS scheduling for the LTE UL.

We denote by A the collection of all sets of contiguousRBs, where A � PðMÞ. PðMÞ represents the power set ofM, i.e., the collection of all subsets of M and 8a 2 A,a ¼ fc; cþ 1; . . . ; cþ lg, 1 � c � cþ l � m. For a; b 2 A, we


Fig. 2. A feasible FDPS scheduling for the LTE uplink. m ¼ 12; n ¼ 5.The shadow (user, RB) pairs denote the RB-to-user assignment of thefeasible schedule. Note that the RBs assigned to each user must becontiguous in frequency domain.

call a intersects b if a \ b 6¼ ;. 8a 2 A, we use headðaÞ todenote the smallest RB of a, and tailðaÞ the largest RB of a.The Boolean variable xai is used to indicate whether or notthe set of contiguous RBs a is assigned to user i. User i getsthe set a 2 A if and only if xai ¼ 1.2

We define the profit function as follows:

pða; iÞ : A�N ! IR;�02

where pða; iÞ indicates the profit gained by assigning a 2 Ato user i in a feasible schedule (in one TTI).

The profit function pða; iÞ is a general term and mightvary with different schedulers. We can exploit it to modelvarious specific scheduling algorithms. For example,pða; iÞ ¼

Pc2a �

ci is the Proportional Fair scheduling objec-

tive studied in [12], where �ci is the PF metric value thatuser i has on RB c. In addition, we can, respectively, expressthe three objective functions in [14] as

pða; iÞ ¼ Qi

Xc2a

Rci ; ð3Þ

pða; iÞ ¼ Qi min

�Qi;Xc2a

Rci

�; ð4Þ

pða; iÞ ¼ ðQiÞ2 ��

max

�0; Qi �

Xc2a

Rci

��2

; ð5Þ

where Qi is the queue size for user i at the beginning of eachTTI, and Rc

i is the data rate for user i, RB c in this TTI. Thesethree objective functions combine throughput maximizationand queue stability together. In Section 7, we specify pða; iÞas (3) to incorporate the queue length and channel qualityinformation, and thus formalize a specific FDPS problem toanalyze the stability of our proposed scheduling algorithm.

3.2 LTE UL FDPS

We consider a general FDPS problem for the LTE uplink.We are given an uplink system with m RBs and n users. Inone time slot, for each set of contiguous RBs a 2 A and eachuser i, we have a profit pða; iÞ. Our goal is to schedule thesystem for this time slot. In other words, we intend to findthe most advantageous way to assign an a 2 A to user i sothat the total profit is maximized. The LTE UL FDPSproblem is formalized as the following combinatorialoptimization problem:

maxX

ða;iÞ2A�Npða; iÞ � xai

subject to :

for each RB c 2M :X

i2N;a:c2axai � 1;

for each user i 2 N :Xa2A

xai � 1;

for i 2 N; a 2 A : xai 2 f0; 1g:

ð6Þ

The first constraint shows that every RB is assigned to atmost one user, and the second constraint ensures that each

user can get no more than one set of contiguous RBs. In fact,LTE UL FDPS aims to find a subset of A�N that maximizesits total profit in a time slot, according to the schedulingpolicy specified in the profit function. Problem (6) is abinary integer programming and it is not hard to find thatthe PF-FDPS problem studied in [12] is a special case of (6).

4 HARDNESS RESULTS

4.1 Hardness of (6)

It is not difficult to show that LTE UL FDPS is NP-hard. Leeet al. [12] have shown that the LTE UL PF-FDPS problem, aspecial case of LTE UL FDPS, is NP-hard. Furthermore, it isstraightforward to reduce the LTE UL PF-FDPS problem toLTE UL FDPS (by simply setting pða; iÞ ¼

Pc2a �

ci ). Thus,

LTE UL FDPS is NP-hard.Furthermore, we have a stronger result here.

Theorem 3. LTE UL FDPS is MAX SNP-hard, and hence, itdoes not have a PTAS assuming P 6¼ NP .

Proof. We prove this theorem by presenting an L-reductionfrom JISP2 to LTE UL FDPS.

Assume that X is an instance of JISP2. X has n jobs,

and we denote them by J1; J2; . . . ; Jn. Job Ji has two

intervals, namely ½sð1Þi ; fð1Þi Þ and ½sð2Þi ; f

ð2Þi Þ. s

ðjÞi and f

ðjÞi are

integers, and fðjÞi > s

ðjÞi , j ¼ 1; 2, i ¼ 1; 2; . . . ; n.

Now, we construct function R. RðXÞ is defined as

follows: Let m ¼ maxi¼1;2; j¼1;...nffðjÞi g � 1. RðXÞ is an LTE

uplink system, which has n users and m RBs. N ¼f1; 2; . . .ng is the set of active users. A is the set of all

contiguous RBs such that 8a 2 A, a ¼ fi; iþ 1; . . . ; iþ lg;1 � i � iþ l � m. User i corresponds to job Ji, i ¼ 1;

2; . . . ; n, and the profit function p : A�N ! IR�0 is

defined as follows:

pða; iÞ ¼1 if a ¼

�sðjÞi ; s

ðjÞi þ 1; . . . ; f

ðjÞi � 1

�;

j ¼ 1; 2;0 otherwise:

8<:

Since sðjÞi and f

ðjÞi are integers, and f

ðjÞi > s

ðjÞi , j ¼ 1; 2,

i ¼ 1; 2; . . . ; n, the profit function p is well defined.Next, we construct function S. Assume that s is a

feasible solution of RðXÞ. s can be written as fðai; uiÞ ji ¼ 1; 2; . . . ; k; ai 2 A; ui 2 Ng. We define psupp ¼ fða; iÞ jpða; iÞ ¼ 1; ða; iÞ 2 A�Ng. Let s0 ¼ s \ psupp. Obviously,s0 is also a feasible solution of RðXÞ such that eachelement of s0 has a positive profit. That is, in s0, each userat most selects one set of contiguous RBs and no two aisappearing in s0 intersect.

According to the definition of the profit function p,8ðai; uiÞ 2 s0, job Jui is associated with an interval½headðaiÞ; tailðaiÞ þ 1Þ in X. Therefore, SðsÞ is defined as

SðsÞ ¼ f ½headðaiÞ; tailðaiÞ þ 1Þ; Juið Þ j ðai; uiÞ 2 s0g:

We use the pair ð½headðaiÞ; tailðaiÞ þ 1Þ; JuiÞ to denotethat in SðsÞ, job Jui selects the interval ½headðaiÞ; tailðaiÞ þ1Þ. Since s0 is a feasible solution of RðXÞ, correspondinglyin SðsÞ, each job at most selects one interval, and no twoselected intervals intersect. So, SðsÞ is a feasible solutionof X. Thus, function S is well defined.


2. IR�0 denotes the nonnegative real number set, i.e., pða; iÞ � 0 for alla 2 A; i 2 N .

Now, we check (1). Assume that s0 is the optimalsolution of RðXÞ, and define s00 ¼ s0 \ psupp. Apparently,we have

VALðs00Þ ¼ VALðs0Þ ¼ OPT ðRðXÞÞ;

where VALðs00Þ and V ALðs0Þ denote the values of

solutions s00 and s0, respectively. We know that Sðs0Þ is

a feasible solution of X, and we have

VALðs00Þ ¼ VALðSðs0ÞÞ � OPT ðXÞ:

Thus, we have OPT ðRðXÞÞ � OPT ðXÞ. So, (1) holds

(� ¼ 1).Then, we check (2). Let s be any feasible solution of

RðXÞ. Obviously, we have

VALðsÞ ¼ VALðs0Þ ¼ V ALðSðsÞÞ; ð7Þ

where s0 ¼ s \ psupp. Assume that r is the optimal

solution of X, and r ¼ fð½sui ; fuiÞ; JuiÞ; i ¼ 1; 2; . . . kg. In

RðXÞ, we can correspondingly construct s ¼ fðfsui ; sui þ1; . . . ; fui � 1g; uiÞ; i ¼ 1; 2; . . . ; kg. In s, each user is

assigned to at most one set of contiguous RBs, and any

two sets of contiguous RBs do not intersect. So, s is a

feasible solution of RðXÞ. Moreover, since for RðXÞ,

pðfsui ; sui þ 1; . . . ; fui � 1g; uiÞ ¼ 1; i ¼ 1; 2 . . . ; k;

we have

OPT ðRðXÞÞ � V ALðsÞ ¼ V ALðrÞ ¼ OPT ðXÞ: ð8Þ

Combining (7) and (8), we have

OPT ðXÞ � VALðSðsÞÞ � OPT ðRðXÞÞ � VALðsÞ:

So, (2) holds (� ¼ 1).Thus, ðR;SÞ is an L-reduction from JISP2 to LTE UL

FDPS. Since JISP2 is MAX SNP-hard, LTE UL FDPS isalso MAX SNP-hard and it does not have a PTAS unlessP ¼ NP . tu

Theorem 3 is somewhat devastating, because the non-

existence of PTAS implies that for some constant > 0,

there are no polynomial time ð1þ Þ-approximation algo-

rithms for LTE UL FDPS unless P ¼ NP . That is to say, we

could at most hope for approximation algorithms that have

constant approximation ratios.3

4.2 The Size of Search Space of (6)

Despite the fact that the LTE UL FDPS problem is MAX

SNP-hard, one may still be tempted to find the optimal

solution by an exhaustive search, because this approach does

not consume much computation power when the search

space is small, and enumerating all feasible schedules is

sufficient to find the optimal schedule.In the following, we calculate the number of feasible

schedules for the exhaustive search in an uplink system,

which has m RBs and n active users. Assume that in a

feasible schedule, k out of n users are assigned to

contiguous RBs. These k sets of contiguous RBs are denoted

as a1; . . . ; ak such that 1 � headða1Þ � tailða1Þ < headða2Þ �tailða2Þ � � � < headðakÞ � tailðakÞ � m. So,

1 � headða1Þ < tailða1Þ þ 1 < headða2Þ þ 1 < tailða2Þþ 2 � � � < headðakÞ þ k� 1 < tailðakÞ þ k � mþ k:

Thus, there is a 1-1 correspondence between the number of

choices of k sets of contiguous RBs and the number of 2k

integers fbi; i ¼ 1; 2; . . . ; 2kg such that 1 � b1 < � � � < b2k �mþ k. So, the number of choices of k sets of contiguous RBs

is ðmþk2k Þ. After assigning k users to k sets of contiguous RBs,

we have ðmþk2k Þ � n!ðn�kÞ! feasible schedules in which k active

users are assigned to contiguous RBs. So, the total number

of feasible schedules is

Xnk¼0

mþ k2k

� �� n!

ðn� kÞ! >mþ n

2n

� � n! ð9Þ

In practical systems, the running time of an exhaustive

search is evidently unacceptable for one TTI (in LTE, 1TTI =

1 ms). In other words, the hardness result and the giant

search space of (6) imply that approximation algorithms

computable in polynomial time with guaranteed perfor-

mance is indispensable for the LTE UL FDPS problem. In

the subsequent sections, we present two polynomial time

approximation algorithms. The first one is intuitive and

easy to follow, but its performance declines slightly as the

number of active users grows. The second one is relatively

delicate, but achieves a better performance. The second

algorithm is derived from the local ratio technique [15].

5 A GREEDY STRATEGY-BASED ALGORITHM

Here, we present our first approximation algorithm. The

main idea of the heuristic algorithm is to divide the LTE

UL FDPS problem into several subproblems according to

the profit, and then apply a greedy method to each

subproblem. We prove that this algorithm has an approx-

imation ratio of OðlnnÞ, where n is the number of active

users in a cell.

5.1 A Special Scheduling Problem

We start from considering a special scheduling problem:

maxX

ða;iÞ 2 C�A�Nxai

subject to:

for each RB c 2M:X

ða;iÞjða;iÞ2C;c2axai � 1;

for each user i 2 N:Xða;iÞ2C

xai � 1;

for each ða; iÞ 2 C: xai 2 f0; 1g:

ð10Þ

In this problem, a user i may not choose a set of

contiguous RBs arbitrarily, but from C, a collection of

legitimate sets of ða; iÞ pairs. All pairs of one user and one

of his legitimate sets of contiguous RBs constitute the

set C, which is a subset of A�N . In addition, pða; iÞ ¼ 1;

8ða; iÞ 2 C. That is, the goal of this problem is to schedule

as many users as possible in a time slot. Problem (10) also


3. For PTAS, we can have a ð1þ �Þ-approximation algorithm computablein polynomial time for any � > 0.

complies with the constraints on users and RBs of the LTE

UL FDPS problem.JISPk problem studied in [20] is very similar to (10).

Based on [20], we provide a greedy algorithm for (10).

headðaÞ and tailðaÞ are the smallest and the largest RBs in

a 2 A, respectively.The main idea of Algorithm 1 is straightforward. It takes

as an input C, a collection of admissible pairs of one user

and one set of contiguous RBs. The algorithm iteratively

selects an ða; iÞ, which has the smallest tailðaÞ until C is

empty. Algorithm 1 outputs G, a feasible schedule of (10).

The value of Algorithm 1 is jGj. According to a similar

Theorem in [20], we can obtain the following theorem. The

prove procedure is the same as that in [20], which is

therefore omitted here.

Algorithm 1. GREEDY.1: input C2: G ;, C C3: while C 6¼ ; do

4: ða; iÞ arg minða;iÞ2CðtailðaÞÞ // break ties arbitrary

5: C Cnfða; iÞ j i ¼ i; or headðaÞ � tailðaÞg6: G G [ fða; iÞg7: end while

8: return G

Theorem 4. Algorithm 1 is a 2-approximation algorithm for (10).

Moreover, it can be shown that the approximation ratio of 2

is tight.

5.2 The Weighted Version of (10)

Then, we consider the weighted version of (10). We

associate each pair ða; iÞ 2 C with profit pða; iÞ, and change

the objective function to

maxXða;iÞ2C

pða; iÞ � xai : ð11Þ

The constraints of (11) are the same as those of (10). This

problem is analogous to LTE UL FDPS except that a

feasible schedule is selected from a subset C � A�N , not

from A�N itself.We consider the following procedure that provides a

feasible solution for (11). For a subset C � A�N , we

ignore the profit function and run Algorithm 1, and then

get a feasible schedule G. Then, we compute WG ¼Pða;iÞ2G pða; iÞ. We denote the optimal value of (11) by

W -OPT ðCÞ. The following lemma establishes the relation

between WG and W -OPT ðCÞ.Lemma 4. If 8ða; iÞ 2 C, 0 < mp � pða; iÞ �Mp, then

W -OPT ðCÞ � 2Mp

mp�WG.

Proof. If C ¼ ;, then the lemma holds vacuously. Otherwise,

we denote the optimal value of (10) by S-OPT ðCÞ, then

W -OPT ðCÞ �Mp � S-OPT ðCÞ. According to Theorem 4,

we have S-OPT ðCÞ � 2 � jGj, so W -OPT ðCÞ � 2Mp � jGj.According to the proposed procedure, we have

WG � mp � jGj. Combining the above two equations,

we finally get W -OPT ðCÞ � 2Mp

mp�WG. tu

5.3 The Approximation Algorithm

Taking Algorithm 1 as the subroutine, our first approxima-tion algorithm for the LTE UL FDPS problem is stated asfollows (Algorithm 2).

Algorithm 2. GREEDY-BASED (G-B for short).

1: input m, n, p

2: pmax maxða;iÞ2A�Npða; iÞ3: partition A�N into kþ 1 subsets: S0; S1; . . . ; Sk such

that S0 ¼ fða; iÞ j pða; iÞ � pmaxn g and Sj ¼ fða; iÞ j

�j�1�pmaxn < pða; iÞ � �j�pmax

n g for j � 1

4: // � > 1 is a constant and k ¼ lnnln�

�. The specification

of � will be discussed in the proof of Theorem 55: for j ¼ 1 to k do

6: Gj GREEDY ðSjÞ7: WGj

Pða;iÞ2Gj

pða; iÞ8: end for

9: j arg max1� j� kðWGjÞ10: return Gj and WGj

Algorithm 2 takes m;n; p as the input, where m is thenumber of RBs, n is the number of active users, andp : A�N ! IR�0. It first partitions all ða; iÞ pairs into kþ 1subsets fSj; j ¼ 0; 1; . . . ; kg according to their profits. ForSk; k � 1, Algorithm 2 invokes Algorithm 1 to obtain afeasible schedule. Thus Algorithm 2 obtains k feasibleschedules, then it chooses the schedule that has the largesttotal profit as the output.

Theorem 5. Algorithm 2 is an OðlnnÞ-approximation algorithm

for LTE UL FDPS, where n is the number of active users in

a cell.

Proof. For any instance S of (6), we denote the optimal

value by OPT ðSÞ and the return value of Algorithm 2 by

WGj . First, we show that OPT ðSÞWGj

� �þ 2�ln� lnn.

In Algorithm 2, all ða; iÞ pairs are partitioned into kþ1 subsets fSj; j ¼ 0; 1; . . . ; kg. Each Sj with correspondingprofits can be regarded as an instance of (11), whoseoptimal value is written as W -OPT ðSjÞ. Then, it isobvious that

OPT ðSÞ �Xkj¼0

W -OPT ðSjÞ: ð12Þ

For j � 1, �j�1�pmaxn < pða; iÞ � �j�pmax

n ; 8ða; iÞ 2 Sj. So,Lemma 4 indicates that W -OPT ðSjÞ � 2� �WGj; j � 1.For j ¼ 0, because the number of users is n, the secondconstraint of (6) tells us that W -OPT ðS0Þ � n � pmaxn ¼pmax. Combining the above two equations, we turn (12)into OPT ðSÞ � pmax þ 2�

Pkj¼1 WGj. Note that fða; iÞ j

pða; iÞ ¼ pmaxg � Sk, so Sk 6¼ ;. Thus, WGk � �k�1�pmaxn . So,

we have

OPT ðSÞ � n

�k�1WGk þ 2�

Xkj¼1

WGj:

Since WGj ¼ max1�j�kWGj, we get

OPT ðSÞ � n

�k�1þ 2� � k

� �WGj � �þ 2�

ln�lnn

� ��WGj :

Thus,


OPT ðSÞWGj

� �þ 2�

ln�lnn: ð13Þ

Furthermore, we can specify � in terms of n so that theright side of (13) is minimized. Let

@ �þ 2�ln� lnn

� @�

¼ 0;

then we get 1� 2nðln�Þ2 þ

2nln� ¼ 0. Since � > 1, we have � ¼

expð 2nnþffiffiffiffiffiffiffiffiffiffi2nþn2p Þ. Substituting this expression of � into (13),

we finally get

OPT ðSÞWGj

� exp2 lnn

lnnþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilnnð2þ lnnÞ

p !

�

1þ lnnþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilnnð2þ lnnÞ

p� � 2e � ðlnnþ 1Þ:

Since S is any instance of (6), we conclude thatAlgorithm 2 has an approximation ratio of OðlnnÞ. tu

Fig. 3 depicts the approximation ratio of Algorithm 2 fordifferent numbers of active users. A cell typically hasseveral tens of active users. In Fig. 3, we can find that theapproximation ratio of Algorithm 2 increases slowly as thenumber of active users n grows.

5.4 A Word on Complexity

Algorithm 2 is based on an intuitive idea that utilizes agreedy method. We have shown that the gap between theoptimal value and the value returned by the algorithmwidens gradually as the number of active users nincreases. Here, we show that Algorithm 2 has a smalltime complexity.

We denote byN the size of A�N andN ¼ n � ððm2 Þ þmÞ.We use N j to represent the size of Sj, j � 1. Note thatPk

j � 1N j � N .

Algorithm 2 first finds pmax and partitions A�N intoS0; S1; . . . ; Sk, which requires a traversal of all elements ofA�N . This takes OðN Þ time.

For each Sj; j � 1, Algorithm 2 calls the subroutineAlgorithm 1. Algorithm 1 iteratively seeks an available ða; iÞwith smallest tailðaÞ, which usually requires a sort forða; iÞ 2 Sj based on tailðaÞ. Thus, it can be shown that for Sj,invoking Algorithm 1 takes OðN j lnN jÞ time. Moreover,

Xkj¼1

OðN j lnN jÞ ¼Xkj¼1

OðN j lnNÞ ¼ OðN lnNÞ:

So, the total running time of Algorithm 2 can becalculated as

TG-B ¼ OðN Þ þXkj¼1

OðN j lnN jÞ ¼ OðN lnNÞ

¼ O n �m2 lnðn �m2ÞÞ ¼ Oðn �m2ðlnnþ lnmÞ�

:

ð14Þ

6 A LOCAL RATIO TECHNIQUE-BASED ALGORITHM

The basic idea of Algorithm 2 is intuitive; however, theperformance is not so satisfying when the number ofactive users grows (Fig. 3). In this section, we introduce amore delicate approximation algorithm based on the localratio technique [15], which achieves a constant approx-imation ratio of 2. The basic idea of local-ratio is similarto the Dynamic Programming, which aims at solvingcomplex problems by breaking them down into simplersubproblems.

6.1 The Algorithm

The approximation algorithm is listed as Algorithm 3. Ittakes the number of RBs (m), the number of active users(n), and the profit function (p : A�N ! IR�0) as theinput. It outputs a feasible schedule S and its total profitW . Algorithm 3 first iterates from 1 through m to findcandidate ða; iÞs for S. In each loop, the algorithm triesto find the best candidate ða; iÞ (in the meaning of profit),and uses a stack S to store it. After the iteration,Algorithm 3 pops each ða; iÞ from S and adds ða; iÞ toS if it is valid. Finally, the algorithm generates a feasibleschedule S.

Algorithm 3. A Local Ratio Technique Based Algorithm

(L-R for short).

1: input m, n, p

2: p0 p, S ; // S is a stack

3: for j ¼ 1 to m do

4: ða; iÞ arg maxða;iÞ2fða;iÞjtailðaÞ¼jgðp0ða; iÞÞ// break ties arbitrary

5: if p0ða; iÞ � 0 then

6: continue

7: end if

8: S:pushðða; iÞÞ9: for each ða; iÞ such that i ¼ i or a intersects a do

10: if p0ða; iÞ > 0 then

11: p0ða; iÞ p0ða; iÞ � p0ða; iÞ12: end if

13: end for

14: end for

15: S0 ;16: while S 6¼ ; do

17: ða; iÞ S:popðÞ18: if S0 [ fða; iÞg is a valid schedule then

19: // a valid schedule means that S0 [ fða; iÞgshould meet the constraints of (6)

20: S0 S0 [ fða; iÞg21: end if

22: end while

23: S S0, W Pða;iÞ2S pða; iÞ

24: return S and W


Fig. 3. Approximation ratio of Algorithm 2.

6.2 The Approximation Ratio

In this section, we will prove that Algorithm 3 has anapproximation ratio of 2, and that this approximation ratiois tight. To obtain this result, we first introduce Lemma 5,which is an instance of the Local Ratio Theorem [15]. Then,we prove that Algorithm 3 is a 2-approximation algorithmfor LTE UL FDPS. Finally, we use an example to show thatthe approximation ratio of 2 is tight.

Lemma 5. Consider an uplink system which has m RBs andn active users. Let p, p1, p2 be profit functions such thatpða; iÞ ¼ p1ða; iÞ þ p2ða; iÞ; 8ða; iÞ 2 A�N . Let S, S1 , andS2 be optimal schedules for p, p2, and p3, respectively. Assumethat S is a feasible schedule such that r �

Pða;iÞ2S p1ða; iÞ �P

ða;iÞ2S1p1ða; iÞ and

r �Xða;iÞ2S

p2ða; iÞ �Xða;iÞ2S

2

p2ða; iÞ;

is a constant. Then, we have

r �Xða;iÞ2S

pða; iÞ �Xða;iÞ2S

pða; iÞ:

Proof. Note thatXða;iÞ2S

pða; iÞ ¼Xða;iÞ2S

p1ða; iÞ þXða;iÞ2S

p2ða; iÞ

�Xða;iÞ2S

1


2

p2ða; iÞ:

On the other hand,Xða;iÞ2S

1


2

p2ða; iÞ � r �Xða;iÞ2S

p1ða; iÞ

þ r �Xða;iÞ2S

p2ða; iÞ ¼ r �Xða;iÞ2S

pða; iÞ:

Thus, the lemma is proved. tuNote that in Lemma 5, the profit functions p, p1, and p2

need not be nonnegative functions. That is to say, the profitfunction can take a negative value for some ða; iÞ 2 A�N .In Theorem 6, we will prove that Algorithm 3 has a constantapproximation ratio of 2. The proof of this result relies onthe observation that Lemma 5 applies to Algorithm 3 andwe will use Lemma 5 inductively in the proof.

Theorem 6. Algorithm 3 is a 2-approximation algorithm for LTEUL FDPS.

Proof. We first introduce some necessary notations anddefinitions.

After the outermost for-loop is finished, the stack Scan be represented as [mj¼1Sj, where each jSjj � 1. If inthe jth for-loop, the algorithm adds some ðaj ; ij Þ to S,then Sj ¼ fðaj ; ij Þg. Otherwise, Sj ¼ ;.

Correspondingly, the outermost for-loop iteratively

generates a series of profit functions ðpðjÞ1 ; pðjÞ2 Þ; j ¼ 1; . . . ;

m. For j � 1, if Sj ¼ fðaj ; ij Þg,

pðjÞ1 ða; iÞ ¼

pðj�1Þ2 ðaj ; ij Þ � 1IR>0

�pðj�1Þ2 ða; iÞ

i ¼ ij or aintersects aj ;

0 otherwise;

8<:

and

pðjÞ2 ða; iÞ ¼ p

ðj�1Þ2 ða; iÞ � pðjÞ1 ða; iÞ 8ða; iÞ 2 A�N:

1IR>0ðxÞ is the characteristic function such that 1IR>0ðxÞ ¼1 for x > 0, and 1IR>0ðxÞ ¼ 0 for x � 0.

If Sj ¼ ;, we set pðjÞ1 ða; iÞ ¼ 0 and p

ðjÞ2 ða; iÞ ¼ p

ðj�1Þ2 ða; iÞ.

We let pð0Þ2 ða; iÞ ¼ pða; iÞ and

pð0Þ1 ða; iÞ ¼ 0; 8ða; iÞ 2A�N:

Thus, we have pðjÞ1 ða; iÞ þ p

ðjÞ2 ða; iÞ ¼ p

ðj�1Þ2 ða; iÞ; j � 1;

ða; iÞ 2 A�N . According to the algorithm, it can beshown that for j � 1,

pðjÞ2 ða; iÞ � 0; 8tailðaÞ � j;

and that

8ða; iÞ 2 A�N; pðjÞ2 ða; iÞ � pðkÞ2 ða; iÞ; j � k:

In addition, the while-loop equivalently generates aseries of Sj ; j ¼ 0; . . . ;m, where S0 ¼ S, Sm ¼ ;, andSj � [

jþ1k¼mSk. S

j is regarded as the value of variable S0

after the algorithm tries to add [jþ1k¼mSk to S0. So, it is

obvious that

Sjþ1 � Sj � Sjþ1 [ Sjþ1; j ¼ 0; . . . ;m� 1:

We denote by WðjÞopt the total profit of the optimal

schedule for pðjÞ2 . If the optimal schedule is empty, we set

WðjÞopt ¼ 0. In particular, W

ð0Þopt is the optimal value of

(6). We define W ðjÞ ¼Pða;iÞ2Sj

pðjÞ2 ða; iÞ. If Sj ¼ ;,

W ðjÞ ¼ 0. In particular, W ð0Þ ¼W . So, what we are going

to prove is Wð0Þopt � 2W ð0Þ ¼ 2W .

In the following, we will prove by induction thatWðjÞopt � 2W ðjÞ, j ¼ m;m� 1; . . . ; 1; 0. When j ¼ 0, we

obtain the desired result. The mathematical inductionstarts from m down to 0.

The Basis. When j ¼ m, pðmÞða; iÞ � 0, tailðaÞ � m. That

is, pðmÞða; iÞ � 0, 8ða; iÞ 2 A�N . So, WðmÞopt ¼ 0. Since

Sm ¼ ;, WðmÞopt � 2W ðmÞ holds vacuously.

The Inductive Step. Assume that WðjÞopt � 2W ðjÞ, j � m.

we denote by VðjÞopt the total profit of the optimal schedule

for pðjÞ1 .

If Sj ¼ ;, then pðjÞ2 ða; iÞ ¼ p

ðj�1Þ2 ða; iÞ, and p

ðjÞ1 ða; iÞ ¼ 0.

So, Wðj�1Þopt ¼W ðjÞ

opt. Since Sj � Sj�1 � Sj [ Sj, Sj�1 ¼ Sj .

Thus,

W ðj�1Þ ¼X

ða;iÞ2Sj�1

pðj�1Þ2 ða; iÞ ¼

Xða;iÞ2Sj

pðjÞ2 ða; iÞ ¼W ðjÞ:

Then, we get

Wðj�1Þopt ¼W ðjÞ

opt � 2W ðjÞ ¼ 2W ðj�1Þ:

Otherwise, Sj ¼ fðaj ; ij Þg. According to the algorithm,pðjÞ2 ðaj ; ij Þ ¼ 0. Because Sj � Sj�1 � Sj [ Sj,

2X

ða;iÞ2Sj�1

pðjÞ2 ða; iÞ ¼ 2

Xða;iÞ2Sj

pðjÞ2 ða; iÞ

¼ 2W ðjÞ �W ðjÞopt:

ð15Þ


On the other hand, Sj�1 contains at least one elementða0; i0Þ such that i0 ¼ ij or a0 intersects a. According to thealgorithm, fða0; i0Þg ¼ Sk; k � j. Thus,

pðj�1Þ2 ða0; i0Þ � pðk�1Þ

2 ða0; i0Þ > 0:

So, pðjÞ1 ða0; i0Þ ¼ p

ðj�1Þ2 ðaj ; ij Þ. In this case,X

ða;iÞ2Sj�1

pðjÞ1 ða; iÞ � p

ðjÞ1 ða0; i0Þ ¼ p

ðj�1Þ2 ðaj ; ij Þ:

Then, we derive an upper bound for VðjÞopt . We define

Suppp1¼ fða; iÞ j pðjÞ1 ða; iÞ > 0g. Suppp1

contains ða; iÞssuch that i ¼ ij or a intersects aj . Moreover, p

ðj�1Þ2 ða; iÞ �

0; tailðaÞ � j� 1, so for ða; iÞ 2 Suppp1, tailðaÞ � j. Since

tailðaj Þ ¼ j, ða; iÞ 2 Suppp1such that i 6¼ ij must intersects

aj at RB j. So, the optimal solution for pðjÞ1 at most have two

ða; iÞs 2 Suppp1: for one ða; iÞ, i ¼ ij , and for another ða; iÞ,

a intersects aj . That is, VðjÞopt � 2p

ðj�1Þ2 ðaj ; ij Þ. Thus,

2X

ða;iÞ2Sj�1

pðjÞ1 ða; iÞ � 2p

ðj�1Þ2 ðaj ; ij Þ � V

ðjÞopt : ð16Þ

Since pðj�1Þ2 ða; iÞ ¼ pðjÞ1 ða; iÞ þ p

ðjÞ2 ða; iÞ, according to

(15) and (16), Lemma 5 indicates that

2W ðj�1Þ ¼ 2X

ða;iÞ2Sj�1

pðj�1Þ2 ða; iÞ �W ðj�1Þ

opt :

Since both the basis and the inductive step have beenproved, it has now been proved by mathematicalinduction that W

ð0Þopt � 2W . So, Algorithm 3 has an

approximation ratio of 2. tu

Moreover, we find an LTE UL FDPS problem that shows

that the approximation ratio of Algorithm 3 is tight.

Proposition 1. The approximation ratio of 2 is tight.

Proof. In the LTE UL FDPS problem of Table 1, m ¼ n ¼ 2,

and 0 < � < 1. The optimal solution is fðf2g; 1Þ; ðf1g; 2Þg,and the optimal value is 2� �. However, for this instance

of LTE UL FDPS, Algorithm 3 returns S ¼ fðf1g; 1Þgand W ¼ 1. So, the approximation ratio � 2��

1 ; 80 <� < 1. When �! 0, 2� �! 2. Thus, the approximation

ratio is tight. tu

6.3 A Word on Complexity

We have shown that Algorithm 3 has a constant approx-

imation ratio, and here we analyze the complexity of the

algorithm. In the jth for loop, the number of ða; iÞs such that

tailðaÞ ¼ j is m � j. So, to obtain ðaj ; ij Þ, the algorithm at leastaccesses m � j ða; iÞs. In addition, since in the jth for loop,p0ða; iÞ � 0, tailðaÞ � j � 1, the number of ða; iÞs suchthat p0ða; iÞ is changed is at least ðm2 Þ þm�

jðj�1Þ2 þ ðn � 1Þ �

j � ðm� jþ 1Þ. So, the outermost for loop takes

Xmj¼1

m � jþ m

2

� þmþ ðn� 1Þ � j � ðm� jþ 1Þ � jðj� 1Þ

2

� �

¼ 1

6�mþm3 þ 5mnþ 6m2nþm3n�

¼ Oðm3nÞ

running time. Since S contains at most m elements, thewhile loop takes OðmÞ time. Thus, the running time ofAlgorithm 3 is

TL-R ¼ Oðm3nÞ: ð17Þ

7 STABILITY ANALYSIS

The assumption of an infinitely backlogged model inscheduling analysis is conventional in many literatures.Nevertheless, in real networks, packets are generated foreach user in accordance with a stochastic arrival processwith respect to applications, which may result in theinstability of queues. Therefore, it is preferable to takesystem utility maximization and queuing stability intoaccount simultaneously when we design scheduling algo-rithms. Since the L-R Algorithm already achieves a fairlygood performance, we now investigate its stability. In thissection, we formalize a specific LTE UL FDPS problem,which is a special case of the general problem (6), and theL-R Algorithm can also be applied with the sameapproximation ratio of 2. Similar to the methodology usedin [14], we utilize the Lyapunov Drift to prove the queuingstability of the L-R Algorithm in LTE UL system.

7.1 A Specific Problem

We consider a specific problem that incorporates the queuelength and channel quality information in the LTE UL. Inthis case, we allocate each set of contiguous RBs a 2 A touser i to maximize

Pi2NP

a2AP

a:c2a QiXai R

ci in each TTI.

The new optimization problem is formalized as follows:

maxXi2N

Xa2A

Xa:c2a

QiXai R

ci ;

subject to :

for each RB c 2M:X

i2N;a:c2aXai � 1;

for each user i 2 N :Xa2A

Xai � 1;

8i 2 N; 8a 2 A: Xai 2 f0; 1g:

ð18Þ

To solve the problem (18), we actually intend to find themost advantageous way to assign an a 2 A to user i so as tomaximize the throughput and keep the queue lengthsbounded. The specific problem can be also considered as amulticarrier variant of the well-known MaxWeight Matchproblem. The first constraint of (18) shows that every RB isassigned to at most one user. The second constraint ensuresthat each user can get no more than one set of contiguousRBs. Especially, for any user i,

Pa2A X

ai ¼ 0 means the user


TABLE 1An LTE UL FDPS Problem which

Approaches the Approximation Ratio

will have no chance to send data in this TTI. Now, we provethe L-R Algorithm solves (18) with the same approximationratio of 2 as it does in (6).

Lemma 6. The L-R Algorithm also solves the specific LTE ULFDPS problem (18) in a polynomial runtime with a constantapproximation ratio of 2.

Proof. We can utilize the profit function to express theobjective of (18) like (3):

pða; iÞ ¼ Qi

Xa:c2a

Rci :

Here, the profit pða; iÞ is obtained as long as thecontiguous set of RBs a is assigned to user i. In otherwords, the profit value exactly equals to the product ofqueue length Qi and the total data rate

Pa:c2a R

ci .

Evidently, the LTE UL FDPS problem (18) is a specialcase of (6). The L-R Algorithm can also be applied here tosolve it with the same approximation performance, andthus we prove the lemma. tu

7.2 Necessary Definitions

Before we conduct the stability analysis, the necessarydefinitions are presented. The definition of ð!; �Þ-admissiblefor single-carrier system is introduced in [27]. We extendthat definition to the LTE UL multicarrier scenario.

Definition 2. We say that a system is ð!; �Þ-admissible if theadversary has a valid schedule set of Y a

i ðtÞ 2 f0; 1g such thatin any window ½t0; t0 þ !� 1�, for the arrival process Aiðt0Þand channel rate Rc

iðtÞ we have

Xt0þ!�1

t0¼t0Aiðt0Þ � ð1� �Þ

Xt0þ!�1

t¼t0

Xa2A

Xa:c2a

Y ai ðtÞRc

iðtÞ 8i; ð19Þ

where ! 2 Zþ, � < 1, andP

i;a:c2a Yai ðtÞ ¼ 1; 8t; i; c.

Informally, an algorithm is said to be stable if it keeps thequeue sizes bounded whenever this is achievable [14]. Moreformally, we define the queuing stability of schedulingalgorithm for LTE uplink as follows:

Definition 3. An scheduling algorithm for LTE uplink is stable ifit keeps the queue lengths bounded for any ð!; �Þ-admissiblesystems.

Suppose that a feasible scheduler Y ensures the arrivalprocess A and channel rate R satisfy the relation of (19),the queue will not increase steeply and its length can bebounded. Accordingly, the ð!; �Þ-admissible explicitlydescribes the basic propositions of some LTE UL systems,which will be stably scheduled by the feasible FDPSalgorithms.

The procedure of stability proof follows a standardtechnique utilized in [14]. In this paper, the procedure iscalled Lyapunov drift analysis, which is similarly conductedin [28] and [25] as well. The fundamental analysis consistsin showing the Lyapunov function has a negative driftwhen the queue lengths are sufficiently large. Here,the Lyapunov drift is computed by Lðtþ !Þ � LðtÞ, whereLðtÞ ¼

PiðQiðtÞÞ2 is the Lyapunov function. For reference

simplified, we use some notations to represent different

algorithms in the following contents. Namely, X, X, and

Y denote the L-R Algorithm, the Optimal Algorithm and any

other Algorithm for Problem (18), respectively. More

specifically, X, X, and Y comprise the Boolean variables

Xai , Xai and Y a

i , respectively.To investigate the stable conditions for the L-R Algorithm

X, it is acceptable to assume that the Optimal Algorithm X

is stable for any ð!; �Þ-admissible LTE UL system, where

! 2 Zþ, � < 1. Afterwards, we figure out the propositions of

ð!0; �0Þ-admissible LTE UL systems, which will be stably

scheduled by the L-R Algorithm X. Moreover, the para-

meters ð!0; �0Þ are calculated and expressed by ð!; �Þ, which

are supposed as known parameters.

7.3 Stability Analysis

Now, we perform the Lyapunov drift analysis for the L-R

Algorithm, which approximately solves the LTE UL FDPS

problem (6) and (18). Some necessary and reasonable

assumptions for LTE UL system used in the stability

analysis are listed below:

1. The channel rates are bounded in a finite set, i.e.,Rsup is the supremum of these rates while Rinf isthe infimum.

2. The arrival process is also bounded, i.e., Asup is thesupremum while Ainf is the infimum.

3. The queue sizes among all the n users within sometime interval ðt0; t0 þ !� 1Þ is bounded, i.e., Qsup isthe supremum while Qinf is the infimum.

The stability conclusions with the values of ð!0; �0Þ are

demonstrated in Theorem 7.

Theorem 7. The L-R Algorithm X is stable for any ð!0; �0Þ-admissible LTE UL systems as long as 8i; c, the user rates

RciðtÞ cannot be zero for arbitrarily long periods, where

1

2� �0 � 1þ ðn� 1ÞAinf

B4� 2ð1� �ÞB3Q

sup

B4Qinf; ð20Þ

and

!0 � !: ð21Þ

Proof. It takes two steps to prove Theorem 7. We start with

showing that in any ð!0; �0Þ-admissible LTE UL systems,

once the queue lengths are sufficiently large then the

Lyapunov function has a negative drift. Afterwards, we

suppose that the parameters ð!; �Þ are already known for

the stable X, and hence ð!0; �0Þ of X can be calculated

and expressed by them.First of all, the fundamental relation among queue

lengths in different TTI yields (22), where ½x�þ denotesmaxð0; xÞ,

Qiðt0 þ !0Þ ¼ Qiðt0Þ þXt0þ!0�1

t¼t0AiðtÞ

"

�Xt0þ!0�1

t¼t0

Xa2A

Xa:c2a

Xai ðtÞRc

iðtÞ#þ

8i 2 N;ð22Þ

which can be rewritten as


Qiðt0 þ !0Þ � Qiðt0Þ þXt0þ!0�1

t¼t0AiðtÞ

�Xt0þ!0�1

t¼t0

Xa2A

Xa:c2a

Xai ðtÞRc

iðtÞ 8i 2 N:ð23Þ

To simplify the expressions, the abbreviations of

notationsP

i ,P

a;c , andP

t stand forP

i2N ,P

a2AP

a:c2a ,

andPt0þ!0�1

t¼t0 , respectively. Then, (23) is substituted into

the Lyapunov drift and we have

Lðt0 þ !0Þ � Lðt0Þ¼Xi

Qiðt0 þ !0Þ2 �Xi

Qiðt0Þ2

¼ 2Xi

�Xt

AiðtÞ�2

þ 2Xi

�Xt

Xa;c

Xai ðtÞRc

iðtÞ�2

þ 2Xi

Qiðt0Þ�X

t

AiðtÞ �Xt

Xa;c

Xai ðtÞRc

iðtÞ�

�Xi

�Xt

AiðtÞ�2

�Xi

�Xt

Xa;c

Xai ðtÞRc

iðtÞ�2

� 2Xi

Xt

AiðtÞ�X

t

Xa;c

Xai ðtÞRc

iðtÞ�;

where the last three terms form a perfect square, and

hence we obtain

Lðt0 þ !0Þ � Lðt0Þ

� 2Xi

�Xt

AiðtÞ�2

þ 2Xi

�Xt

Xa;c

Xai ðtÞRc

iðtÞ�2

þ 2Xi

Qiðt0Þ�X

t

AiðtÞ �Xt

Xa;c

Xai ðtÞRc

iðtÞ�:

ð24Þ

Since the system is ð!0; �0Þ-admissible, the totalamount of arriving data for each user is upper bounded(see (19)) by a function ofRsup,m, !0, and �0. In particular,

Xt0þ!0�1

t¼t0AiðtÞ � ð1� �0Þ

Xt0þ!0�1

t¼t0

Xa;c

Xai ðtÞRc

iðtÞ

� ð1� �0Þ!0mRsup:

Consequently, the first term of the right side of (24)has an upper bound while the second term is alsobounded by a function of Rsup, m, !0, n, and �0. Wedenote B1 as the upper bound of the first two terms, i.e.,

B1 ¼ 2nðð1� �0Þ2 þ 1Þð!0mRsupÞ2:

As for the third term, provided that t0 � t � t0 þ !0,the following inequality holds obviously:

Qiðt0Þ � QiðtÞ þ !0mRsup:

Hence, we have


� B1 þ 2!0mRsupXi

Xt

�AiðtÞ �

Xa;c

Xai ðtÞRc

iðtÞ�

þ 2Xi

QiðtÞ�X

t

AiðtÞ �Xt

Xa;c

Xai ðtÞRc

iðtÞ�:

ð25Þ

Similarly, the second term of the above expression canbe upper bounded by a function of Rsup, m, n, and !0,which equals to B2:

B2 ¼ �2n�0ð!0mRsupÞ2:

Besides, the objective values of (18) under differentalgorithms form the succeeding inequality:

2Xt;i;a;c

QiðtÞXai ðtÞRc

iðtÞ �Xt;i;a;c

QiðtÞXai ðtÞRciðtÞ

�Xt;i;a;c

QiðtÞY ai ðtÞRc

iðtÞ:ð26Þ

Again, both ð!0; �0Þ-admissible proposition and (26)are applied together. In consequence, (25) can berewritten as


� B1 þB2 þ 2Xi

QiðtÞ�X

t

AiðtÞ �Xt

Xa;c

Xai ðtÞRc

iðtÞ�

� B1 þB2 þ 2Xi

Xt

Xa;c

QiðtÞRciðtÞ�

ð1� �0ÞY ai ðtÞ �Xa

i ðtÞ

� B1 þB2 þ 2Xi;t;a;c

ð1� 2�0ÞQiðtÞXai ðtÞRc

iðtÞ:

ð27Þ

Evidently, the parameter �0 needs to satisfy (20) (i.e.,12 � �0) to keep the third term of (27) nonpositive.Suppose j ¼ argminiQiðtÞ and QminðtÞ ¼ QjðtÞ, then(28) is hence derived:

Lðt0 þ !0Þ � Lðt0Þ� B1 þB2 þ 2ð1� 2�0Þ

Xi;t;a;c

QiðtÞXai ðtÞRc

iðtÞ

� B1 þB2 þ 2ð1� 2�0ÞRinfmXt

QminðtÞ:ð28Þ

Once the queue lengths are sufficiently large, thepotential function Lðt0Þ decreases, which implies thestability of the L-R Algorithm X. So far the stability isproved, we turn to compute the key parameters ð!0; �0Þ,and use ð!; �Þ of X to express them.

Obviously, when � < 1, inequality (26) yields

2ð1� �ÞXt;i;a;c

QiðtÞXai ðtÞRc

iðtÞ

� ð1� �ÞXt;i;a;c

QiðtÞXai ðtÞRciðtÞ:

ð29Þ

Combining the ð!; �Þ-admissible proposition for thestable X, the bounded queue lengths and (29), we have

2ð1� �ÞQsupXt;i;a;c

Xai ðtÞRc

iðtÞ

� 2ð1� �ÞXt;i;a;c

QiðtÞXai ðtÞRc

iðtÞ

� ð1� �ÞXt;i;a;c

QiðtÞXai ðtÞRciðtÞ

� QinfXt;i;a;c

AiðtÞ:


Likewise,P

t;i;a;c Xai ðtÞRc

iðtÞ can be upper bounded bya function (denoted by B3) of m, n, !, and Rsup. 8i 2 N ,the total arriving data of user i has a lower bound, andthus we obtain

2ð1� �ÞQsupB3 � QinfXt;i;a;c

AiðtÞ

� QinfXt¼t0þ!�1

t¼t0AiðtÞ þQinfðn� 1ÞAinf :

ð30Þ

As long as we set !0 � ! (see (21)), (30) can berewritten as follows:

Xt¼t0þ!0�1

t¼t0AiðtÞ �

Xt¼t0þ!�1

t¼t0AiðtÞ

� 2ð1� �ÞB3Qsup

Qinf� ðn� 1ÞAinf :

So far, we find out an upper bound forPt¼t0þ!0�1

t¼t0 AiðtÞ.To guarantee the LTE UL system is ð!0; �0Þ-admissibleso that X schedules stably, we require (19) holds forð!0; �0Þ, i.e.,

Xt0þ!0�1

t¼t0AiðtÞ � ð1� �0Þ

Xt0þ!0�1

t¼t0

Xa2A

Xa:c2a

Xai ðtÞRc

iðtÞ 8i: ð31Þ

Moreover, the right part of (31) can be intuitively lowerbounded by ð1� �0Þ!0mR

inf . We use B4 to represent!0mR

inf . Consequently, (31) holds provided that

2ð1� �ÞB3Qsup

Qinf� ðn� 1ÞAinf � ð1� �0Þ!0mR

inf ;

where �0 can be eventually derived as

�0 � 1þ ðn� 1ÞAinf

B4� 2ð1� �ÞB3Q

sup

B4Qinf:

In other words, the LTE UL system can be consideredas ð!0; �0Þ-admissible in case ð!0; �0Þ satisfy (20) and (21),respectively. In this kind of cellular systems, the L-RAlgorithm schedules stably. Hence, we prove the theoremand complete the stability analysis. tu

8 PERFORMANCE EVALUATION

In this section, we utilize the LTE UL simulator [29], which

is an open platform for academic research. On that platform,

we deploy the Algorithm 3 and perform simulations for the

Local Ratio (L-R) scheduler (Algorithm 3) as well as otherwell-known schedulers, including Max-Min fairness, BestCQI, Proportional-Fair and Round Robin (RR). To imple-ment the Algorithm 3, the profit function is specified as thecurrent data rate, namely,

pða; iÞ ¼Xc2a

Rci ;

where Rci is the average data rate of user i on RB c within

the present TTI.We carry out five similar simulations and calculate the

average cell throughput4 for each scheduler. The results areprovided with the average values of a 10,000 TTIs durationfor each number of UEs. The operational frequency is2 GHz while the frequency bandwidth is 20 MHz. Twokinds of user speed are set, i.e., the walking mode (5 km/h)and the vehicular mode (50 km/h). As illustrated in Figs. 4and 5, the RR and Best CQI (that always schedules userwith the best channel quality) algorithms are the lower andupper bounds of all schedulers in terms of systemperformance (represented by cell throughput in this case),respectively. Along with the increasing number of users,the average throughput of all schedulers trends to decreaseslightly. The throughput of the L-R scheduler is betweenthe PF and the upper bound (Fig. 4), which indicates itsgood performance. When the user speed is accelerated to50 km/h, due to serious channel fading, the average cellthroughput of all schedulers are degraded (Fig. 5). How-ever, in this mode, the L-R scheduler still achieves betterperformance than the PF scheduler.

To investigate the stability of the L-R scheduler, wecompute a stability region based on Theorem 7. It isassumed that the Optimal Algorithm for problem (6) isstable for any (1-0)-admissible LTE system, namely ! ¼ 1,� ¼ 1

2 . In this case, to hold (20) and (21), we consider12 � 1þ ðn�1ÞAinf

B4� B3Q

sup

B4Qinf and !0 ¼ 1. In other words, the2-approximation L-R Algorithm is stable as long as Qsup

is set,

Qsup � B4

2B3þ ðn� 1ÞAinf

B3: ð32Þ

Furthermore, we present simulation results of theaverage queue length of the L-R scheduler when differentarrival rates are set. The queue length are sampled and


Fig. 4. The average throughput of all the UEs. (User speed is 5 km/h.) Fig. 5. The average throughput of all the UEs. (User speed is 50 km/h.)

4. The cell throughput is the total throughput of all the uploading UEsthat a eNodeB covers.

averaged in a period of 100 TTIs, i.e., 0.1 s. The arrivalrates A are also set 50/60/70/80 Mbps. Each rate lasts for10,000 TTIs. According to (32), we can plot a stability regionfor the L-R scheduler. Theoretically, the queue length willreside in this region without overflow when a certainarrival rate is set for the L-R scheduler. The stability region(the dash line) and the simulation results (the solid line) arecombined in Fig. 6. It indicates that the queues are stablewhen the arrival rates are relatively small. However, whenthe arrival rate is 80 Mbps, the average queue length isincreasing steeply and exceeds the stability region, whichimplies overflow and the instability of the schedulingalgorithm. In other words, the L-R Algorithm is stable forLTE UL system within a certain admissible region of thearrival rate.

9 CONCLUSION

In this paper, we consider a general FDPS problem for the

LTE uplink. Our formulation of this problem applies to

many scheduling policies. We prove that the LTE UL FDPS

problem is MAX SNP-hard, which implies that the problem

is hard to approximate. We propose two approximation

algorithms for this scheduling problem, both of which are

computable in polynomial time. The first algorithm is based

on a simple greedy method, and it has an approximation

ratio of OðlnnÞ, where n is the number of active users in the

cell. The second L-R Algorithm is more delicate, which

though achieves a constant approximation ratio of 2.Furthermore, we perform Lyapunov drift analysis to

investigate the stability of the L-R Algorithm. We firstspecify the general profit function to incorporate queuelength and channel quality information. The specificproblem (18) is a special case of (6). Accordingly, the L-RAlgorithm can be also applied with the same approxima-tion ratio of 2. Subsequently, we show the Lyapunov driftfor problem (18) is negative once the queue lengths aresufficiently large. Finally, the L-R Algorithm is proved to bestable for any ð!0; �0Þ-admissible LTE UL system. Thesimulation results indicate good performance of the LocalRatio scheduler.

ACKNOWLEDGMENTS

This work was supported in part by the National GrandFundamental Research Program of China (973) under Grant

No. 2010CB328105, the National Natural Science Founda-

tion of China (NSFC) under Grant No. 61225011, and the

National Science and Technology Major Project of China

(NSTMP) under Grant No. 2011ZX03002-002-02. A pre-

liminary version [1] of this paper appeared in the

Proceedings of IEEE INFOCOM 2010, San Diego, California,

14-19 March, 2010.

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Fengyuan Ren received the BA and MScdegrees in automatic control from NorthwesternPolytechnic University, China, in 1993 and 1996,respectively, and the PhD degree in computerscience from Northwestern Polytechnic Univer-sity in December 1999. He is a professor in theDepartment of Computer Science and Technol-ogy at Tsinghua University, Beijing, China. From2000 to 2001, he was with the ElectronicEngineering Department at Tsinghua University

as a postdoctoral researcher. In January 2002, he moved to theComputer Science and Technology Department at Tsinghua University.His research interests include network traffic management, control in/over computer networks, wireless networks and wireless sensornetworks. He has (co)authored more than 80 international journal andconference papers. He is a member of the IEEE and has served as atechnical program committee member and local arrangement chair forvarious IEEE and ACM international conferences.

Yinsheng Xu received the BA degree fromSchool of Software, Beijing Institute of Technol-ogy, in 2008 and is currently working toward thePhD degree in the Department of ComputerScience and Technology, Tsinghua University,China. His current research interests includeresource scheduling and congestion control inmobile Internet. In addition, he has completedsome work on real-time transmission in wirelesssensor networks.

Hongkun Yang received the BS and MSdegrees in computer science in 2007 and2010, respectively, both from the Departmentof Computer Science and Technology, TsinghuaUniversity, China. He is currently working towardthe PhD degree in the Computer ScienceDepartment at the University of Texas at Austin.His research interests include wireless networksand network security.

Jiao Zhang received the BA degree incomputer science and technology from theBeijing University of Posts and Telecommuni-cation in 2008, and is currently working towardthe PhD degree in the Department of Com-puter Science and Technology, Tsinghua Uni-versity, China. Her recent research focuses ontraffic management in data center networks.She has also done some work on dataaggregation and energy-efficient routing in

wireless sensor networks.

Chuang Lin received the PhD degree incomputer science from Tsinghua University,China, in 1994. He is a professor in theDepartment of Computer Science and Technol-ogy at Tsinghua University, Beijing, China. He isan Honorary visiting professor at the Universityof Bradford, United Kingdom. His current re-search interests include computer networks,performance evaluation, network security analy-sis, and Petri net theory and its applications. He

has published more than 300 papers in research journals and IEEEconference proceedings in these areas and has published four books.He served as the technical program vice chair of the the 10th IEEEWorkshop on Future Trends of Distributed Computing Systems(FTDCS 2004) and general chair of the ACM SIGCOMM Asia workshop2005 and 2010 IEEE International Workshop on Quality of Service(IWQoS 2010). He is an associate editor for the IEEE Transactions onVehicular Technology and an area editor of Computer Networks andthe Journal of Parallel and Distributed Computing. He is a seniormember of the IEEE and the Chinese Delegate in TC6 of IFIP.

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